Green forms and the arithmetic Siegel–Weil formula

Abstract

We construct natural Green forms for special cycles in orthogonal and unitary Shimura varieties, in all codimensions, and, for compact Shimura varieties of type \(\mathrm {O}(p,2)\) and \(\mathrm {U}(p,1)\), we show that the resulting local archimedean height pairings are related to special values of derivatives of Siegel Eisentein series. A conjecture put forward by Kudla relates these derivatives to arithmetic intersections of special cycles, and our results settle the part of his conjecture involving local archimedean heights.

Notes

Acknowledgements

This work was done while L.G. was at the University of Toronto and IHES and S.S. was at the University of Manitoba; the authors thank these institutions for providing excellent working conditions. An early draft, with a full proof of the main identity for non-degenerate Fourier coefficients, was circulated and posted online in January 2018; we are grateful to Daniel Disegni, Gerard Freixas i Montplet, Stephen Kudla and Shouwu Zhang for comments on it, and other helpful conversations. We also thank the referee for their thorough reading and insightful comments and suggestions. L.G. acknowledges financial support from the ERC AAMOT Advanced Grant. S.S. acknowledges financial support from NSERC.